Tracing Cyclic Homology Pairings Under Twisting of Graded Algebras

TRACINGCYCLICHOMOLOGY

PAIRINGSUNDERTWISTINGOFSayan Chakraborty∗ Makoto Yamashita†

GRADEDALGEBRASSeptember 11, 2017

Abstract We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss–Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.

Cyclic homology theory as initiated by Tsygan [Tsy83] and Connes [Con85] provides a natural exten- sionintroduction of de Rham theory for manifolds to objects of noncommutative geometry, which are represented by noncommutative algebras. The main subject of this paper is the cyclic cohomology for deformation of algebras graded by some discrete group G, with deformation parameter given by exponential of a 2-cocycle on G. This deformation scheme has many good analogies with the theory of (formal) deformation quantization of Poisson manifolds, but at the same time gives a concrete example of strict deformation in the sense of Rieffel, meaning that we can specialize the deformation parameter to concrete numbers and represent the deformed algebra on Hilbert spaces. The most fundamental example 2 in this setting is the celebrated noncommutative torus Tθ, which is represented by a twisted group algebra of Z2, and is also a deformation quantization of T2 with respect to the translation invariant Poisson structure. Our goal is to understand: (1) the cyclic cohomology of such algebras as abstract linear spaces, and (2) the pairing with K-groups / cyclic homology groups for deformed algebras, in terms of those for the original algebra and the cohomology of G. For the case of the group algebra of G = Z2 (which leads to the noncommutative torus), these were carried out in [Rie78,PV 80, Ell84, Con85], but beyond the commutative (or ﬁnite, in which case this deformation is trivial) case, the full understanding is arXiv:1709.03337v1 [math.QA] 11 Sep 2017 limited to only a handful of cases. A notable case is the crossed product of Z2 by a ﬁnite subgroup of SL2(Z), where relatively comprehensive description of K-groups, cyclic cohomology groups, and their pairings for twisted group algebras are known [Wal95, Wal00,BW 07, Qud14, Qud16]. Let us also mention that another interesting work from a geometric viewpoint is carried out by Mathai [Mat06] for the standard traces of twisted group algebras for many interesting classes of groups. Regarding the point (1) above, usual computation of cyclic cohomology starts with the computation of Hochschild cohomology groups, which serve as analogue of the spaces of differential forms. Once this is done, the cyclic cohomology groups can be obtained using the map induced by Rinehart–Connes

*: [email protected], Westfalische¨ Wilhelms-Universitat¨ Munster¨ †: [email protected], Ochanomizu University

1 operator, playing the role of the exterior derivative. However, while the end result of computation, 2 particularly the periodic cyclic cohomology groups tend to be invariant under deformation of algebra structures, the intermediate Hochschild cohomology is very sensitive to deformation. This is already apparent in the case of noncommutative torus, and there is little hint for the more general case of deformation of graded algebras by 2-cocycles, even just for the twisted group algebras. Our approach, instead, is based on the Gauss–Manin connection for periodic cyclic (co)homology introduced by Getzler [Get93]. The monodromy for this connection, if it makes sense, will give a natural isomorphism between the periodic cyclic cohomology of the deformed algebra and that of the original one. This plays a particularly powerful role in the setting of Poisson deformation quantization, and nicely explains the relation between the Poisson homology of a Poisson manifold and the cyclic homology of the deformed algebra. Motivated by this, our viewpoint could be phrased as follows: the group algebra of G represents a quantum group, which is (up to “Poincare´ dual”) homotopically modeled on the classifying spaces for centralizers of torsion elements of G.A 2-cocycle ω0 on G represents an (invariant) “Poisson bivector” on this quantum group. A G-grading on an algebra corresponds to an action of this quantum group, by which we can transport the “Poisson bivector” and related structures. A new challenge in the setting of strict deformation is that the Gauss–Manin connection is intrinsi- cally defined on the infinite dimensional space of periodic cyclic (co)chains, and there is no general theorem to integrate it. Developing the second named author’s previous work [Yam12] on “invariant” cyclic cocycles, we solve this issue by identifying the action of connection operator up to chain homotopy by the cup product action of group cohomology. As it turns out, this formalism also fits well with recent developments in the operator algebraic K C([0 1]) -theory of such algebras [ELPW10, Yam11], where one considers the√ , -algebra whose fiber at t ∈ [0, 1] is the deformation with respect to the rescaled 2-cocycle et −1ω0 . If G satisfies the Baum– Connes conjecture with coefficients, the evaluation at each t induces isomorphism of K-groups, which basically solves the problem (2). On the purely algebraic side, the image of “C -sections” in periodic cyclic homology are flat with respect to the Gauss–Manin connection. This allows∞ us to express the pairing of K-groups and cyclic cohomology of the deformed algebra in terms of the original data and the action of group cohomology. As an application, we look at the twisted group algebras of crystallographic groups. The K-theory of untwisted algebra was studied in detail by Luck¨ and his collaborators [LS00,DL13,LL12], and our result give a description of cyclic cohomology and its pairing with K-groups for twisted group C∗-algebras. The paper is organized as follows. In Section 1, we recall the basic terminology and prepare nota- tions. In Section 2, exploiting the comparison of Hochschild cohomology of group algebra and group cohomology [Bur85], we construct traces on twisted group algebras which are supported on the conjugacy classes of torsion elements. This leads to isomorphism between the cyclic cohomology groups of general 2-cocycle deformation of graded algebras over such conjugacy classes, solving the problem (1). Section 3 is the main part of the paper, where we compute the monodromy of the Gauss–Manin connection with respect to the identification established in Section 2. In Section 4, we establish K-theoretic comparison between Frechet´ algebraic models and operator algebraic models, as a basis to compute the pairing of operator algebraic K-theory and cyclic cohomology of “smooth” algebras. In Section 5, we apply our general theory to twisted algebras of crystallographic groups. Acknowledgements. The authors would like to thank Wolfgang Luck¨ and Ryszard Nest for fruitful comments. They also thank the following programmes / grants for their support which enabled collaboration for this paper: Simons - Fundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund; The Isaac Newton Institute for Mathematical Sciences for the programme Operator algebras: subfactors and their applications. This work was supported by: EPSRC grant number EP/K032208/1 and DFG (SFB 878). M.Y. thanks the operator algebra group at University of Munster¨ for their hospitality during his stay, and Yasu Kawahigashi for financial support (JSPS KAKENHI Grant Number 15H02056). preliminaries 3

1 preliminaries 1.1Let usNoncommutative ﬁrst review the calculus fundamental concepts in cyclic homology theory. Basic references are [Cun04, Tsy04]. ⊗n+1 Let A be a unital C-algebra, and put Cn(A) = A . Recall that the Hochschild differential Cn(A) → Cn−1(A) is given by

n−1 i n b(a0 ⊗ · · · ⊗ an) = (−1) a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an + (−1) ana0 ⊗ a1 ⊗ · · · ⊗ an−1,(1.1) i=0 X while the Rinehart–Connes differential Cn(A) → Cn+1(A) is

n ni B(a0 ⊗ · · · ⊗ an) = (−1) 1 ⊗ an+1−i ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ an−i i=0 X n + (−1) an+1−i ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ an−i ⊗ 1 .

⊗n The reduced Hochschild complex of A is given by C¯ n(A) = A ⊗ (A) , where A = A/C. The kernel of quotient map Cn(A) → C¯ n(A) is stable under b, and is contractible (in the sense of complex). Thus, ∗ (C¯ ∗, b) also computes the Hochschild homology of A, and its linear dual complex C¯ (A) computes the Hochschild cohomology. On this model B is given by

n ni B: C¯ n(A) → C¯ n+1(A), a0 ⊗ · · · ⊗ an → (−1) 1 ⊗ ai ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ ai−1. i=0 X We also denote the transpose maps on C¯ ∗(A) by b and B. n ⊗n The A-valued Hochschild n-cochains are by deﬁnition elements of C (A; A) = HomC(A , A). The direct sum with degree shift C∗−1(A; A) is a graded pre-Lie algebra by partial composition D ◦ D0, and the product map of A regarded as µ ∈ C2(A; A) gives the Hochschild differential δ(D) = [µ, D] by the graded commutator. Given such a cochain D ∈ Cm(A; A), following [Get93] (but in the convention of [Yam12]) we consider the operators

BD(a0 ⊗ · · · ⊗ an) n(j+1)+m(k−j) = (−1) 1 ⊗ aj+1 ⊗ · · · ⊗ D(ak+1 ⊗ · · · ⊗ ak+m) ⊗ · · · ⊗ an ⊗ a0 ⊗ · · · ⊗ aj. 0 j k 6X6 6n−m and mn bD(a0 ⊗ · · · ⊗ an) = (−1) D(an−m+1 ⊗ · · · ⊗ an) ⊗ a0 ⊗ · · · ⊗ an−m. acting on C∗(A). Then ιD = bD − BD is called the interior product by D. Another important operation is the Lie derivative

n−m m(j+1) LD(a0 ⊗ ... ⊗ an) = (−1) a0 ⊗ · · · ⊗ D(aj+1 ⊗ · · · ⊗ aj+m) ⊗ · · · ⊗ an j=0 X n n(n−j) + (−1) D(aj+1 ⊗ · · · ⊗ a0 ⊗ a1 ⊗ · · · ⊗ aj+m−n−1) ⊗ aj+m−n ⊗ · · · ⊗ aj. j=n−m+1 X n Let us denote by C¯ (A; A) the space of cochains with D(a1 ⊗ · · · ⊗ an) = 0 whenever ai = 1 for some i. This is a subcomplex for δ, and ιD and LD are operators on C¯ ∗(A) when D is in this space. We then have the following Cartan homotopy formula relating these two kinds of operators. Proposition 1.1 ([Rin63; Get93; Yam12, Proposition 4]). When1.2D Standard∈ C¯ m(A complex; A), we for have group cohomology 4

[b − B, ιD] = LD − ιδ(D), where the left hand side is the graded commutator in End(C¯ ∗(A)).

Let us recall the formalism of Gauss–Manin connection on periodic cyclic (co)homology [Get93]. Sup- pose (µt)t∈I is a ‘smooth’ family of associative product structures parametrized by I = [0, 1] with common multiplicative unit on A. We then consider Γ (I; A) which is, as a linear space, the union of I A smooth functions from into ﬁnite dimensional subspaces∞ of . The product structure is given point- 0 0 0 wise by µt, that is (f ∗ f )(t) = µt(f(t), f (t)) for f, f ∈ Γ (I; A) (so we require a regularity condition µ on t to make this well-deﬁned). ∞ Then ∂t is a Hochschild 1-cochain of Γ (I; A) which is not a derivation unless µt is constant in t. δ(∂ ) = [µ ∂ ] Note that t , t is up to sign equal∞ to the time derivative of product structure. Hence the Cartan homotopy formula says

[b − B, ι∂t ] = L∂t + ιm˙ t . ¯ as operators on C∗(Γ (I; A)). Note also that L∂t is the natural extension of ∂t to chains by the Leibniz rule. ∞

G M = / × 1.2Let Standardbe a discrete complex group, for group and cohomologybe a commutative group (R, C, T R Z, or C for our purposes). The group cohomology H∗(G; M) is equal to the cohomology of the standard complex (C∗(G; M), d), where Cn(G; M) = Map(Gn, M) and d: Cn(G; M) → Cn+1(G; M) is given by

dφ(g1, ... , gn+1) = φ(g2, ... , gn+1) n i n+1 + (−1) φ(g1, ... , gigi+1, ... , gn+1) + (−1) φ(g1, ... , gn).(1.2) i=1 X We formally put C0(G; M) = Map(∗, M) = M and d to be the zero map from C0(G; M) to C1(G; M). Consequently, H0(G; M) = M and the 2-cocycles are 2-point functions satisfying

φ(h, k) + φ(g, hk) = φ(g, h) + φ(gh, k).

If G is a finite group, one has Hn(G; C) = 0 for n > 0. This is a consequence of the identification ∗ ∗ H (G; C) ' ExtG(C, C) and the semisimplicity of G-modules, but more concretely we can use Shapiro’s 1 lemma type argument: for example, given any 2-cocycle ω0, one defines ψ ∈ C (G; C) by 1 ψ(g) = ω (h, g). |G| 0 h G X∈

Then one can verify dψ(g, h) = ω0(g, h) by concrete computation using the 2-cocycle identity. ∗ ∗ n Obviously, C (G) = C (G; C) is the dual complex of C∗(G) = (C[G ])n=0, endowed with differential d: C (G) → C (G) C∗(G) n+1 n analogous to that of . ∞ A group cochain φ ∈ Cn(G) is normalized if

φ(g1, ... , gi, e, gi+1, ... , gn−1) = 0 (0 6 i 6 n − 1)

and φ(g1, ... , gn) = 0 whenever g1 ··· gn = e. (Note that this requirement is stronger than the usual one found for example in [Bro94].)

Proposition 1.2 ([Kar83;LQ 84, Section 4]). When K is a ﬁeld of characteristic zero and n > 0, any K-valued n-cocycle is cohomologous to a normalized one. While the above does not immediately apply to T- or C×-valued1.3 Cyclic cocycles, cocycles a direct on group argument algebra still 5 shows 2-cocycles with these coefﬁcients can still be normalized. Let us also note the following cyclicity of normalized cocycles.

Proposition 1.3. Let φ be a normalized n-cocycle on G. Then we have

−1 n φ((g1 ··· gn) , g1, ... , gn−1) = (−1) φ(g1, ... , gn).

−1 Proof. Compute the value of 0 = dφ on ((g1 ··· gn) , g1, ... , gn) and use the normalization condition.

[G] 1.3Let usCyclic now cocycles describe on the group cyclic algebra theory for group algebra C , which is essentially from [Bur85] on cyclic homology. Its adaptation to cyclic cohomologies is already explained in [Ji93], but we will later need explicit constructions in this section. For each x ∈ G, let us denote the centralizer of x in G by CG(x), and the conjugacy class of x by AdG(x). x ⊗n+1 Let Cn(C[G]) be the subspace of Cn(C[G]) = C[G] spanned by those g0 ⊗ · · · ⊗ gn such that x g0 ··· gn ∈ AdG(x). The spaces (Cn(C[G]))n=0 give a subcomplex for b and B, which leads to the direct sum decomposition ∞

M x M ∗ x HH∗(C[G]) = H∗(C∗ (C[G]), b), HC∗(C[G]) = H (CC(C∗ (C[G])), b − B).

x∈G/AdG x∈G/AdG

Dualizing this, we also obtain the direct product decomposition

∗ ∗ x 0 ∗ ∗ x 0 HH (C[G]) = H (C∗ (C[G]) , b), HC (C[G]) = H (CC(C∗ (C[G])) , b − B), x∈G/YAdG x∈G/YAdG ∗ ∗ and so on. Let us denote the factors labeled by x by HH (C[G])x and HC (C[G])x. We want to describe them in terms of group cohomology. y y y −1 y From now on let us ﬁx x, and also choose and ﬁx g = gx ∈ G such that (g ) xg = y for each element y in AdG(x). n+1 Let (g0, ... , gn) ∈ G be such that g0 ··· gn ∈ AdG(x). We put

yi = gi ··· gng0 ··· gi−1.

Note that these elements are also in AdG(x).

y y −1 yn y −1 Lemma 1.4. The elements g i gi(g i+1 ) for 0 6 i < n, and g gn(g 0 ) are in CG(x). y y Proof. This follows from direct calculation using xg = g y and yigi = giyi+1.

We next consider the map

x y1 y2 −1 yn y0 −1 Ξ: Cn(C[G]) → Cn(CG(x)), g0 ⊗ · · · ⊗ gn 7→ (g g1(g ) , ... , g gn(g ) ).

x Proposition 1.5. The map Ξ intertwines the Hochschild differential b on (Cn(C[G]))n=0 and d on C∗(CG(x)). Proof. We can do direct comparison for each terms in (1.1) and in (1.2). ∞

Consequently, any cocycle on CG(x) induces a Hochschild cocycle on C[G] which is supported on 0 the conjugacy class of x. For example, the trivial class represented by 1 ∈ C = C (CG(x)) corresponds τx(f) = δ (f) = f(g) C[G] to the trace AdG(x) g∈AdG(x) on . P In fact, the above proposition corresponds to the fact that Ξ is induced1.3 Cyclic by cocycles a map onof cyclic group sets algebrafrom 6 Γ∗(G, x) to B∗(CG(x), x) in the notation of [Lod98]. A cyclic set is a simplicial set

n n n n ((Xn)n=0; (di : Xn → Xn−1)i=0, (si : Xn → Xn+1)i=0 ∞ endowed with tn : Xn → Xn satisfying certain compatibility conditions [Lod98, Chapter 6]. On the x n one hand, Γn(G, x) is the natural bases of Cn(C[G]). On the other, the simplicial set Bn(G) = G is a cyclic set by −1 t(g1, ... , gn) = ((g1 ··· gn) , g1, ··· , gn−1).

The cyclic set B∗(CG(x), x) has the same underlying simplicial structure as B∗(CG(x)), but the cyclic −1 structure t is given by (x(g1 ··· gn) , g1, ··· , gn−1) instead. ∗ ∗ Proposition 1.6. The map Ξ induces an isomorphism HH (C[G])x ' H (CG(x); C).

Proof. This corresponds to the ﬁrst half of [Bur85, Theorem I’]. More concretely, consider a map x Υ: C∗(CG(x)) → C∗ (C[G]) deﬁned by

−1 Υ(g1, ... , gn) = (g1 ··· gn) x ⊗ g1 ⊗ · · · ⊗ gn.

Then, on the one hand, ΞΥ is equal to the identity map on C∗(CG(x)). On the other, ΥΞ can be computed as

y0 y1 −1 y1 y2 −1 yn y0 −1 g0 ⊗ · · · ⊗ gn 7→ g g0(g ) ⊗ g g1(g ) ⊗ ... ⊗ g gn(g ) .

x This map is homotopic to the identity on C∗ (C[G]) as follows [GJ99, Section III.2]. Put

y0 y0 −1 θ0(g0 ⊗ · · · ⊗ gn) = g g0 ⊗ g1 ⊗ · · · ⊗ gn ⊗ (g ) ,

y0 yn −1 yn y0 −1 θ1(g0 ⊗ · · · ⊗ gn) = g g0 ⊗ g1 ⊗ · · · ⊗ gn−1 ⊗ (g ) ⊗ g gn(g ) , ...

y0 y1 −1 y1 y2 −1 yn y0 −1 θn(g0 ⊗ · · · ⊗ gn) = g g0 ⊗ (g ) ⊗ g g1(g ) ⊗ · · · ⊗ g gn(g ) .

Then the alternating sum

n n+i+1 x x hn = (−1) θi : Cn(C[G]) → Cn+1(C[G]) (1.3) i=0 X satisﬁes bhn + hn−1b = ι − ΥΞ. Consequently, Ξ and Υ induce isomorphism of cohomology between the dual complexes.

Lemma 1.7. Let φ be a normalized n-cocycle on CG(x). The Hochschild cocycle φ ◦ Ξ on C[G] descends to C¯ n(C[G]).

Proof. We need to check

φ ◦ Ξ(h0 ⊗ · · · ⊗ hi ⊗ e ⊗ hi+1 ⊗ · · · ⊗ hn−1) = 0 (0 6 i 6 n − 1) whenever h0 ··· hn−1 ∈ AdG(x). Putting (g0, ... , gn) = (h0, ... , hi, e, hi+1, ... , hn−1), one has yi+1 = yi+2 in the above notation and hence the (i + 1)-th component of Ξ(g0, ... , gn) is e. By the normalization condition on φ, we have the required vanishing.

Let NG(x) be the quotient of CG(x) by the subgroup hxi generated by x.

Proposition 1.8. Let φ be a normalized n-cocycle on NG(x). The induced cocycle φ˜ on CG(x) satisﬁes φΞB˜ = 0. y y −1 y y −1 7 Proof. We want to show the vanishing of φ(g j gj(g j+1 ) , ... , g1.3j− Cyclic1 gj−1( cocyclesg j ) ) onfor group each j algebra. Using the cyclicity of Proposition 1.3, this is equal to

n yj yj −1 −1 yj yj+1 −1 yj−2 yj−1 −1 (−1) φ((g yj(g ) ) , g gj(g ) , ... , g gj−2(g ) ).

y y −1 −1 −1 Since (g j yj(g j ) ) is equal to x , which becomes e in NG(x), the normalization condition implies the vanishing of this term.

It follows that normalized cocycles on NG(x) define reduced cyclic cocycles on C[G]. We next want to understand if they survive in the periodic cyclic cohomology. When x is of infinite order, one has n n HC (C[G])x ' H (NG(x); C). Moreover, the Gysin maps associated with the fibration of classifying 1 n n+2 spaces S = B hxi → BCG(x) → BNG(x) defines a linear map f: H (NG(x); C) → H (NG(x); C) n n+2 for each k, which is identified with the S-map HC (C[G])x → HC (C[G])x [Bur85, Section IV; Lod98, Chapter 7]. It follows that the periodic cyclic cohomology class associated with a normalized n-cocycle φ on N (x) only depends on its image in the inductive limit lim Hn+2k(N (x); C). G −→k G Remark 1.9. The above Gysin map can be described as follows. The Hochschild–Serre spectral sequence p q associated with the extension hxi → CG(x) → NG(x) has the E2-page H (NG(x); H (hxi ; C)), which are concentrated on q = 0, 1. The E2-differential

p 1 p+2 0 d2 : H (NG(x); H (hxi ; C)) → H (NG(x); H (hxi ; C))

p p+2 can be identiﬁed with maps H (NG(x); C) → H (NG(x); C) which is exactly the Gysin map. The d2 is also directly described by the cup product with the extension class [And65]. Namely, the central 2 extension hxi → CG(x) → NG(x) corresponds to a class in H (NG(x); hxi). Identifying hxi with Z and 2 embedding it in C, we obtain a class cG;x in H (NG(x); C). The cup product by cG;x is d2. It follows that lim Hn+2k(N (x); C) is trivial if and only if c represents a nilpotent element in the group −→k G G;x cohomology of NG(x).

In particular, if NG(x) has bounded cohomology degree, the conjugacy class of x does not contribute in HP∗(C[G]). Let us denote the class of discrete groups such that any subgroup has bounded cohomology degree by C.

Lemma 1.10. The class C contains ﬁnite groups, ﬁnitely generated commutative groups, free groups, fundamental groups of acyclic manifolds, and is closed under extensions.

Proof. One obtains the statement about extensions from Hochschild–Serre spectral sequence.

When x is of ﬁnite order, the situation is much simpler. First the ﬁniteness of hxi implies Hq(hxi; C) = n n 0 for q > 0, hence the Hochschild–Serre spectral sequence implies H (CG(x); C) ' H (NG(x); C). Usually the following proposition is stated in the homological form, but the proof carries over without change since it is about the simplicial structure of the underlying cyclic set B∗(CG(x); x).

Proposition 1.11 ([Bur85, Section IV; Lod98, Chapter 7]). Suppose that x ∈ G is of ﬁnite order. Then the n n+2 S-map HC (C[G])x → HC (C[G])x is identiﬁed with the canonical inclusion as the direct summand

n n b 2 c b 2 c+1 M n−2k M n−2k+2 H (NG(x); C) → H (NG(x); C), k=0 k=0

hence one has ∗ M k M k HP (C[G])x ' H (NG(x); C) = H (CG(x); C). k≡∗ mod 2 k≡∗ mod 2 1.4 Action of group cohomology on graded algebras 8

1.4Let usAction next of review group ring cohomology and module on graded structure algebras on cyclic cohomology of group algebras and graded algebras, mainly from [Ji93, Yam12]. 0 0 Let A and A be two unital C-algebras. Then the complex C∗(A ⊗ A ) can be regarded as a product 0 0 C∗(A) × C∗(A ) of mixed modules [Lod98, Chapter 4], while C∗(A) ⊗ C∗(A ) is again a mixed mod- ∗ ∗ ules by differentials b ⊗ Id +(−1) Id ⊗b and B ⊗ Id +(−1) Id ⊗B. There is a map C∗(A) ⊗ C∗(A) → 0 C∗(A) × C∗(A ) called the shufﬂe product, which is an quasi-isomorphism for b − B). When φ (resp. ψ) is a cyclic m-cocycle on A (resp. cyclic n-cocycle on A0), the cup product φ ∪ ψ is a cyclic (m + n)-cocycle 0 0 on A ⊗ A which corresponds to the cocycle φ ⊗ ψ in the dual complex of C∗(A) ⊗ C∗(A ). Remark 1.12. There is another approach φ # ψ, which goes through the correspondence between the cyclic cocycles and closed graded traces on differential graded algebras (Ω∗, d: Ω∗ → Ω∗+1) with homomorphism A → Ω0 [Con85]. Then φ # ψ is given by the graded tensor product of the objects involved. When φ (resp. ψ) is a cyclic m-cocycle on A (resp. cyclic n-cocycle on A0), we have 1 1 [φ # ψ] = [φ ∪ ψ] (m + n)! m!n! in HCm+n(A ⊗ A0). L Let A be a G-graded algebra, that is, A decomposes as a direct sum g∈G Ag satisfying 1 ∈ Ae and AgAh ⊂ Agh. This can be encoded as a coaction homomorphism

α: A → A ⊗ C[G], a 7→ a ⊗ g (a ∈ Ag). There is also a direct product decomposition

∗ ∗ HP (A) = HP (A)x, x∈G/YAdG analogous to the case of group algebra. ∗ ∗ ∗ We then have an action of HP (C[G]) on HP (A), given by φ C ψ = α (φ ∪ ψ) in the above notation. A = [G] ∗( [G]) τx = δ In particular, with C , we obtain a ring structure on HP C . The trace AdG(x) is an ∗ ∗ idempotent, and for general G-graded A, it acts on HP (A) as the projection onto HP (A)x. ∗ Proposition 1.13. Let x be of ﬁnite order. The factor HP (C[G])x is a subring, whose product structure is ∗ isomorphic to the cup product on H (CG(x); C). ∗( [G]) δ Proof. That HP C x is a subring follows from the fact that AdG(x) is a central idempotent for the above product. It remains to identify the product structure. ∗ On the one hand, by [Yam12, Corollary 9], the cup product on HC (B∗(NG(x))) coincides with ∗ that of H (NG(x)) under the standard comparison map. On the other, there is a map of cyclic mod- x ules from C∗ (C[G]) to C[B∗(NG(x))] by composing Ξ with the natural surjection C[B∗(CG(x), x)] → C[B∗(NG(x))]. The cup product is natural for map of cyclic modules, hence we obtain the assertion. Remark 1.14. The above proposition is, at least to some extent, known to experts. For example [Ji93] contains a very similar module structure based on φ # ψ. But we feel that some details, such as a k k k−2 precise form of product structure on HC (C[G])x ' H (NG(x); C) ⊕ H (NG(x); C) ⊕ · · · , and the compatibility under the periodicity map S, need some additional elaboration. n n+2 Proposition 1.15. Let x be an element of inﬁnite order in G. The S-map HC (A)x → HC (A)x is equal 2 2 to the cup product by the extension class cG;x ∈ HC (C[G])x = H (NG(x); C). Proof. This is proved in [Ji93, Theorem 6.2] under a different convention of the S-map and cup product (see Remark 1.12). Let us give an alternate shorter proof: the S-map in our convention is the 2 1 n 2 cup product by v ∈ HC (C) characterized by v(1, 1, 1) = − 2 . Identifying HC (A)x ⊗ HC (C) n 2 0 with HC (A)x ⊗ HC (C) ⊗ HC (C[G])x and using the associativity of cup product, we can iden- n S(φ) φ S(1 ∗ ) φ (A) c tify with ∪ H (NG(x)) for ∈ HC x. By Remark1.9, we know that G;x represents S(1 ∗ ) H (NG(x)) . ∗ ∗ 9 Corollary 1.16 (cf. [Ji93, Corollary 6.6]). If HP (C[G])x is trivial, then1.5HP Deformation(A)x is also of trivial. graded algebras

A G ω × 2 1.5Let Deformationbe a -graded of graded algebra, algebras and be a normalized C -valued -cocycle. We can then deﬁne a new associative product on A by a ∗ω b = ω(g1, g2)ab for homogeneous elements a ∈ Ag1 and b ∈ Ag2 . Thanks to the normalization condition on ω, 1A is still the unit for this product. We denote this algebra by Aω, and its element corresponding to a ∈ A by a(ω). There is an embedding of algebra Aω → A ⊗ Cω[G] given by

(ω) (ω) a 7→ a ⊗ λg (a ∈ Ag).(1.4)

When ω is a T-valued normalized 2-cocycle on G, we can represent Cω[G] on `2(G) by bounded operators, as (ω) −1 −1 (λg f)(h) = ω(g, g )f(g h)(f ∈ `2(G)), ∗ ∗ ∗ called the ω-twisted left regular representation. The reduced twisted group C -algebra Cr,ω(G) = Cr(G, ω) is deﬁned as the norm closure of Cω[G] inside B(`2(G)). When A is a G-C∗-algebra, the Banach space

`1(G; A) = f: G → A | kfk1 = kf(g)k < g G X∈ ∞ has a structure of Banach ∗-algebra by convolution product. When ω is a normalized T-valued cocycle, we can again twist the product of `1(G; A) into a Banach ∗-algebra, with respect to the same involution ∗ ∗ map. If G is amenable, Cr,ω(G) is the C -enveloping algebra of `1(G) = `1(G; C) with this product. More generally, if A is the reduced section algebra of a Fell bundle over G (that is, a C∗-algebra with ∗ ∗ injective coaction A → A ⊗ Cr(G)), we can deﬁne a C -algebraic model of Aω on the right Hilbert 2 Ae-module L (A)Ae , so that Aω is again the reduced section algebra of another Fell bundle.

A 1.6Finally,Fr´echetalgebrasa Frechet´ algebra is given by a (possibly nonunital) C-algebra and a sequence of (semi)norms kakm (a ∈ A) for m = 1, 2, 3, ..., such that A is complete with respect to the locally convex topology deﬁned by the kakm, and that the product map A × A → A is (jointly) continuous. Replacing kakm with kak if necessary, we may and do assume that the seminorms are increasing (kak k6m k m 6 kak ). An important condition one may ask on these seminorms is the m-convexity: each kak is mP+1 m submultiplicative, that is kabkm 6 kakm kbkm. If we can characterize the topology of A by seminorms satisfying this, A is then called an m-algebra.

Let ω0(g, h) be a C-valued normalized 2-cocycle on G. We consider the twisted group algebra for × t tω (g,h) t C2 -valuedinducing2-cocycle tracesω (g, h) = one 0 twisted, so that the algebras new product is g ∗t h = ω (g, h)gh. τ = δ [G] t [G] gh (x) We want to ‘deform’ the trace AdG(x) on C to a one on Cω . Note that when ∈ AdG ,

δ (g h) = ωt(g h) δ (h g) = ωt(h g) AdG(x) ∗t , , AdG(x) ∗t , , but we do not necessarily have ωt(g, h) = ωt(h, g). Weinducing want to correcttraces this on twisted situation algebras by setting 10 τx (g) = etξ(g)δ (g) ξ τx ωt AdG(x) for some function . Then ωt becomes a trace if we have

ω0(g, h) + ξ(gh) = ω0(h, g) + ξ(hg) (2.1)

This means that the bilinear extension of ω0(g, h) − ω0(h, g) is the Hochschild coboundary of the a linear extension of ξ. Let us put ω0 (g, h) = ω0(g, h) − ω0(h, g). a Lemma 2.1. The bilinear extension of ω0 represents a Hochschild 1-cocycle on C[G]. Proof. We need to show a a a ω0 (gh, k) − ω0 (g, hk) + ω0 (kg, h) = 0 a whenever ghk ∈ AdG(x). Expanding the deﬁnition of ω0 , this is the same as

ω0(gh, k) + ω0(hk, g) + ω0(kg, h) = ω0(k, gh) + ω0(g, hk) + ω0(h, kg).

Adding ω0(g, h) + ω0(h, k) + ω0(k, g) to both sides and using the cocycle identity, we indeed obtain the equality.

a 1 By Proposition 1.6, ω0 is a Hochschild coboundary if and only if its image in H (CG(x); C) is trivial. 1 1 a Since H (CG(x); C) = Z (CG(x); C), so ω0 is a coboundary if and only if its pullback by Υ vanishes. a a −1 Proposition 2.2. Suppose x is of ﬁnite order. Then the function ω0 ◦ Υ(g) = ω0 (g x, g) on CG(x) is trivial. −1 −1 a −1 a −1 Proof. Using g x = xg and that ω0 is a normalized 2-cocycle, we have ω0 (g x, g) = −ω0 (x, g ). −1 −1 a −1 Note that x and g play the same role in this expression, and x ∈ CG(g ). Thus, ω0 (h, g ) as a −1 1 −1 −1 function in h ∈ CG(g ) is in Z (CG(g ); C) = Hom(CG(g ), C). Since x is of ﬁnite order, it has to vanish on x.

It follows that we have a solution for ξ in (2.1) when x is of ﬁnite order. 2 2 Example 2.3. Suppose we have G = Z o Z3, where the generator 1 of Z3 acts on Z by the matrix

−1 −1 A = . 1 0

In the following we denote elements of G by triples (a, b, i) for a, b ∈ Z and i ∈ Z3. This group has the following conjugacy classes.

2 {(a, b, 0) | (a, b) ∈ Z3.z} (z ∈ Z3\Z ), {(a, b, 1) | a ≡ b mod 3}, {(a, b, 1) | a ≡ b + 1 mod 3}, {(a, b, 1) | a ≡ b + 2, mod3}, {(a, b, 2) | a ≡ b mod 3}, {(a, b, 2) | a ≡ b + 1 mod 3}, {(a, b, 2) | a ≡ b + 2, mod3},

The class of ﬁrst kind with z = (0, 0) contributes by rank 2 in the K0 group, while the ones for nontrivial z do not contribute. The rest of the classes contribute by rank 1 for each. Let us consider the 2-cocycle 2 ω0 on Z given by ω0((a, b), (c, d)) = ad − bc, and extend it to G by

g ω0((a, b, g), (c, d, h)) = ω0((a, b), (c, d) )(a, b, c, d ∈ Z, g, h ∈ Z4). √ 2 θ This represents a generator of H (G; C). We then take ω(g, h) = exp(− −1 2 ω0(g, h)). Let us consider the case x = (0, 0, 1). Then we have √ −1θ ω((a, b, 0), (c, d, 1)) = ω((a, b), (c, d)) = exp − (ad − bc) , 2 √ −1θ ω((c, d, 1), (a, b, 0)) = ω((c, d), (−a − b, a)) = exp − (ca + ad + bd) . 2 11 Thus, in the twisted group algebra Cω[G], we have 2.1 Twisting cyclic cocycles √ (ω) (ω) −1θ (ω) λ λ = exp − (ad − bc) λ ,(2.2) (a,b,0) (c,d,1) 2 (a+c,b+d,1) √ (ω) (ω) −1θ (ω) λ λ = exp − (ca + ad + bd) λ .(2.3) (c,d,1) (a,b,0) 2 (c−a−b,d+a,1)

δ In particular, AdG(x) is not a trace, but a correction like √ −1θ(a2 + ab + b2) τω(λ ) = exp − δ (a, b, 1) (2.4) x (a,b,1) 6 AdG(x) is. Let us relate the extra factor appearing here to the above general discussion. On the one hand, a a a using the chain homotopy hi in (1.3), ω0 b = 0 and ω0 Υ = 0 implies that ξ can be taken as ω0 h0. On y b−a −a−2b y −1 y the other, for y = (a, b, 1) ∈ AdG(x), the element g = ( 3 , 3 , 0) satisﬁes (g ) xg = y. Thus ξ is given by

a2 + ab + b2 ξ(g ) = ωah (g ) = ω (gg0 g , (gg0 )−1) − ω ((gg0 )−1, gg0 g ) = 0 0 0 0 0 0 0 0 3

for g0 = (a, b, 1) with a ≡ b mod 3, and ξ(g0) = 0 otherwise. To check that the functional (2.4) agrees on (2.2) and (2.3), one needs

1 ad − bc + (a + c)2 + (a + c)(b + d) + (b + d)2 = 3 1 ca + ad + bd + (c − a − b)2 + (c − a − b)(a + d) + (a + d)2 , 3 which is indeed the case.

x Let us assume that x is of ﬁnite order, so that we always have ξ satisfying (2.1) and the trace τωt as 2.1 Twisting cyclic cocycles 0 0 x above on Cωt [G]. Then, if τ is a trace on A, τ ⊗ τωt is a trace on A ⊗ Cωt [G]. Let A be G-graded, and (Ω∗, A → Ω0,τ ˜ : Ωn → C) be a closed dg-trace model for some cyclic cocycle on A. Suppose that Ω∗ is also graded over G, and that the homomorphism respects the grading (this ∗ k ∗ is the case for the universal model Ω (A) given by Ω (A) = C¯ k(A)). Then the twisting of Ω by ω is ∗ x a subalgebra of Ω ⊗ Cωt [G] by the inclusion of the form (1.4). Thus we can restrict the traceτ ˜ ⊗ τωt ∗ 0 to Ωωt , call itτ ˜ωt . Since we have an induced homomorphism Aωt → Ωωt , we obtain a dg-algebra trace model ∗ 0 n (Ωωt , Aωt → Ωωt ,τ ˜ωt : Ωωt → C). This way we can deﬁne a map (which depends on the choice of ξ) from the set of cyclic n-cocycles on A to those on Aωt .

Proposition 2.4. Let x be an element of ﬁnite order in G, and φ(a0, ··· , an) be a cyclic cocycle on A, supported on the conjugacy class of x. If we denote a corresponding closed graded trace corresponding to φ by τ˜, the cyclic x cocycle on Aωt corresponding to τ˜ ⊗ τωt is given by

(t) t t tξ(g0···gn) φ (a0, ... , an) = ω (g0, g1) ··· ω (g0 ··· gn−1, gn)e φ(a0, ... , an) when ai are homogeneous elements with ai ∈ Agi , g0 ··· gn ∈ AdG(x). Proof. The correspondence of φ andτ ˜ is given by

φ(a0, ... , an) = τ˜(a0da1 ··· dan). 12 The n-form in Ω(Aωt ) can be computed as monodromy of gauss–manin connection

(ωt) (ωt) (ωt) t t (ωt) a0 da1 ··· dan = ω (g0, g1) ··· ω (g0 ··· gn−1, gn)(a0da1 ··· dan)

(ω) for homogeneous elements ai ∈ Agi . Under the coaction by C[G], the element (a0da1 ··· dan) is (ω) x mapped to (a0da1 ··· dan) ⊗ g0 ··· gn, henceτ ˜ ⊗ τωt gives the formula in the right hand side of the assertion.

−t −tω0(g,h) The inverse cocycle ω (g, h) = e satisﬁes (Aωt )ω−t = A. The associated map of cyclic cocycles for −ξ is the inverse map for the above. It follows that φ 7→ φ(t) is a linear isomorphisms n n from HC (A)x to HC (Aωt )x.

Let ω0 be a C-valued normalized 2-cocycle on G, and x ∈ G be an element of ﬁnite order. Our goal is ∗ 3to computemonodromy the monodromy of of gauss–manin the Gauss–Manin connection connection on the spaces HP (Cωt [G])x.

C (x) 3.1SupposeFinite ﬁrst centralizer that the centralizer G is ﬁnite. Following Section 1.2, we put 1 ψ(g) = ω0(h, g)(g ∈ CG(x)), |CG(x)| h∈XCG(x) so that we have dψ(g, h) = ω0(g, h) on CG(x) × CG(x). In order to analyze traces on the deformed algebras, we want an explicit presentation of ξ on AdG(x) a satisfying ξ(b(g ⊗ h)) = ω0 (g, h). In the notation of the proof of Proposition 1.6,

a a g0 g0 −1 ω0 h0(g0) = ω0 (g g0, (g ) )

a is one candidate, as Proposition 2.2 implies ω0 Υ = 0. For a reason which will become clear soon, we instead take a g0 g0 −1 ψ(x) + ω0 (g g0, (g ) )(g0 ∈ AdG(x)) ξ(g0) = (3.1) 0 otherwise. Note that adding scalar to ξ does not change ξ(b(g ⊗ h)), while etξτx is affected. We then put

y y −1 y −1 y −1 y y −1 ψ(g 1 g1(g 0 ) ) + ω0(g1, (g 0 ) ) − ω0((g 1 ) , g 1 g1(g 0 ) )(g0g1 ∈ AdG(x)) θ(g0, g1) = 0 otherwise (3.2) ⊗2 2 with y0 = g0g1 and y1 = g1g0, as in Section 1.3. If we extend θ to C[G] = C[G ] by linearity, it is a reduced Hochschild cochain by the normalization condition on ω0.

Lemma 3.1. We have Bθ = ξ as a function on C[G].

Proof. As the reduced model gives B(g) = 1 ⊗ g for g ∈ G, we have

g0 −1 g0 −1 θB(g0) = ψ(x) + ω0(g0, (g ) ) − ω0((g ) , x)

g g −1 for g0 ∈ AdG(x). Using x = (g 0 g0)(g 0 ) and the 2-cocycle identity, we indeed obtain ξ(g0).

Let ω0(g, h)gh denote the Hochschild 2-cochain given by the linear extension of g ⊗ h 7→ ω0(g, h)gh. bθ = τx [G3] 3.1 Finite centralizer 13 Lemma 3.2. We have bω0(g,h)gh as a functional on C , where τx(g g g ) = ω (g g )δ (g g g ) bω0(g,h)gh 0, 1, 2 0 1, 2 AdG(x) 0 1 2 .

Proof. Let us keep the notation yi from the proof of Proposition 1.6. Expanding the deﬁnitions, we have

y2 y0 −1 y1 y0 −1 y1 y2 −1 θb(g0 ⊗ g1 ⊗ g2) = ψ(g g2(g ) ) − ψ(g g1g2(g ) ) + ψ(g y1(g ) )

y0 −1 y2 −1 y2 y2 −1 y0 −1 + ω0(g2, (g ) ) − ω0((g ) , g g2(g ) ) − ω0(g1g2, (g ) )

y1 −1 y1 y0 −1 y2 −1 y1 −1 y1 y2 −1 + ω0((g ) , g g1g2(g ) ) + ω0(g1, (g ) ) − ω0((g ) , g g1(g ) ).(3.3)

First, (by Proposition 1.5) the ﬁrst three terms involving ψ are equal to

y1 y2 −1 y2 y0 −1 y1 y2 −1 y2 y0 −1 dψ(g g1(g ) , g g2(g ) ) = ω0(g g1(g ) , g g2(g ) ).(3.4)

y −1 y y −1 Combining this with ω0((g 1 ) , g 1 g1g2(g 0 ) ) in (3.3), we obtain

y1 −1 y1 y2 −1 y2 −1 y2 y0 −1 ω0((g ) , g g1(g ) ) + ω0(g1(g ) , g g2(g ) ).(3.5)

y −1 Note that the ﬁrst term of this cancels with the last term of (3.3). On the other hand, ω0(g2, (g 0 ) ) − y −1 y y −1 ω0((g 2 ) , g 2 g2(g 2 ) ) in (3.3) is equal to

y2 y0 −1 y2 −1 y2 ω0(g g2, (g ) ) − ω0((g ) , g g2).(3.6)

y −1 Combining ω0(g1, (g 2 ) ) in (3.3) and the second term of (3.5), we obtain

y2 −1 y2 y0 −1 y0 −1 ω0((g ) , g g2(g ) ) + ω0(g1, g2(g ) ).(3.7)

y −1 Then the contribution from (3.6) and the ﬁrst term of (3.7) is equal to ω0(g2, (g 0 ) )). Combining this y −1 with −ω0(g1g2, (g 0 ) ) in (3.3) and the second term of (3.7), we indeed obtain the factor ω0(g1, g2) by the cocycle identity.

tω (g,h) Let us consider the family of product structures µt(g, h) = e 0 gh on C[G]. We denote by ω0(g, h)gh the 2-cochain on C[G] characterized by D(g, h) = ω0(g, h)gh. −ξ + ι τx = (b − B)θ Proposition 3.3. We have ω0(g,h)gh . x Proof. Then Lemma 3.2 says that the Hochschild cochain D = ∂tµt satisﬁes bD τ = bθ. By the degree x x reason we have BD τ = 0, so we also have ιDτ = bθ. Combining this with Lemma3.1, we obtain the assertion.

Let A be a G-graded algebra, and let us consider the smooth section algebra Γ (I; Aω∗ ) as in : Γ (I A ∗ ) → A t Section 1.1, together with the evaluation homomorphisms evt ; ω ω . ∞ Theorem 3.4. Let x be an element of G with ﬁnite centralizer. Suppose∞ that A is a G-graded algebra, φ is a cyclic cocycle on A supported on the conjugacy class of x, and let φ(t) denote the induced cyclic cocycle on (t) Aωt corresponding to ξ in (3.1). When c is a (b − B)-cocycle in C¯ ∗(Γ (I; Aω∗ )), the pairing hφ , evt ci is t independent of . ∞ (t) Proof. We want to show that the time derivative of hφ , evt ci is zero. Since the deformations (t) A Aωt and φ φ are additive in t, it is enough to prove this at t = 0. Note also that the embedding (1.4) can be lifted to

Γ (I; Aω∗ ) → A ⊗ Γ (I; Cω∗ [G]).

∞ (t) ∞ x Thus, we may replace Aωt by A ⊗ Cωt [G] and φ by φ ∪ τωt . Moreover, up to the shufﬂe prod- x x uct (which is a quasi-isomorphism) φ ∪ τωt corresponds to φ ⊗ τωt on the mixed complex C∗(A) ⊗ 14 C∗(Cωt [G]). Hence it is enough to show that, if c is a cocycle in C∗(A) ⊗ C∗(Γ 3.2(I; C Generalω∗ [G])) case, the pairing x ∞ h(Id ⊗ evt)(c), φ ⊗ τωt i has zero derivative at t = 0. This amounts to x hev0((Id ⊗∂t)c), φ ⊗ τ i + hev0 c, φ ⊗ ξi = 0,(3.8) where we identify ξ with a Hochschild 0-cochain supported on AdG(x). As explained in Section 1.1, the Cartan homotopy formula for the cochain ∂t implies ( ∂ )c = −( ι )c + ( [b − B ι ])c Id ⊗ t Id ⊗ ω0(g,h)gh Id ⊗ , ∂t .

We have Id ⊗[b − B, ι∂t ] = [b − B, Id ⊗ι∂t ], hence the left hand side of (3.8) is equal to

−h(Id ⊗(b − B)) ev0 c, φ ⊗ θi by Proposition 3.3. Since φ is a (b − B)-cocycle, this is equal to h(b − B) ev0 c, φ ⊗ θi = 0. This shows (t) ∂thφ , evt ci = 0 at t = 0 as required.

Let us now turn to the general case. If x is a torsion element, as remarked before Proposition 1.11, 3.2n General case n H (NG(x); C) is naturally isomorphic to H (CG(x); C). Combined with Proposition 1.2, there is a normalized 2-cocycleω ˜ 0 on NG(x) which is cohomologous to ω0 when pulled back to CG(x). Let ψ denote a function on CG(x) satisfying

ω0(g, h) − ψ(g) − ψ(h) + ψ(gh) = ω˜ 0(g, h)(g, h ∈ CG(x)).(3.9)

On hxi, it can be taken as ord x−1 k 1 k n ψ(x ) = ω0(x , x g), ord x n=0 X which is independent of g ∈ G (use the 2-cocycle condition for xk, xn, and g). Following the previous 0 1 section, we deﬁne ξ ∈ C (C[G])x and θ ∈ C (C[G])x by the same formulas (3.1) and (3.2). Lemma 3.1 holds without change, and Lemma 3.2 becomes the following. bθ = τx + ω Ξ [G3] Lemma 3.5. We have the equality bω0(g,h)gh ˜ 0 ◦ as functionals on C . Proof. The only difference from the proof of Lemma 3.2 is (3.4), which needs to be replaced by

y1 y2 −1 y2 y0 −1 dψ(g g1(g ) , g g2(g ) )

y1 y2 −1 y2 y0 −1 y1 y2 −1 y2 y0 −1 = ω0(g g1(g ) , g g2(g ) ) − ω˜ 0(g g1(g ) , g g2(g ) ).

This gives the extra termω ˜ 0 ◦ Ξ(g0, g1, g2). Consequently, the monodromy of the Gauss–Manin connection becomes the following. −ξ + ι + τx ω = (b − B)θ Proposition 3.6. We have ω0(g,h)gh C 0 . x Proof. The cocycleω ˜ 0 ◦ Ξ represents the cup product action of the cohomology class of ω0 on τ .

The analogue of Theorem 3.4 is the following. Note that the assumption on exp(t[ω0]) is automati- ∗ cally satisﬁed if H (CG(x); C) is bounded in degree, which holds if G is in the class C (see Lemma 1.10). Theorem 3.7. Let x be an element of ﬁnite order in G. Suppose that A is a G-graded algebra, φ is a cyclic (t) cocycle on A supported on the conjugacy class of x, and let φ denote the induced cyclic cocycle on Aωt ∗ constructed with ξ as above. Let us also suppose that exp(t[ω0]) makes sense in H (CG(x); C). When c is a (t) (b − B)-cocycle in C¯ ∗(Γ (I; Aω∗ )), the pairing hφ C exp(−t[ω0]), evt ci is independent of t. ∞ Proof. The proof is parallel to that of Theorem 3.4, but this time we want to check smooth models 15

x x hev0((Id ⊗∂t)c), φ ⊗ τ i + hev0 c, φ ⊗ ξi − hev0 c, φ ⊗ τ C ω0i = 0. x The extra term τ ∪ ω0 exactly corresponds to the one in Proposition 3.6. Corollary 3.8. With the same assumption as above, we have

(t) hφ C exp(t[ω0]), ev0 ci = hφ , evt ci.

We want to consider Frechet´ variants of Γ (I; C[G]) in the previous section. When A is an m-algebra, C (I) ⊗ˆ A m we4 maysmooth take the projective models tensor product∞ , which is again an -algebra (note that tensor product of submultiplicative seminorms are∞ again submultiplicative). It can be identiﬁed with the space C (I; A) of A-valued C -functions on I, or the injective tensor product thanks to the nuclearity C (I) of ∞as a Frechet´ space, cf.∞ [Tre`67, Chapter 44; Phi91, Theorem 2.3]. ∞

`: G [0 ) G `(e) = 0 `(gh) `(g) + `(h) 4.1Let -Schwartz→ , algebrabe a length function on ; a function satisfying and 6 . Let A be a Frechet´ algebra with seminorms kakm and let α: G y A be an action of G which is `-tempered [Sch93∞b]: k ∀m∃C, k, n: kαg(x)km 6 C(`(g) + 1) kxkn . We then put k S1(G; A) = {f: G → A | ∀ k, m: (`(g) + 1) kf(g)km < }.(4.1) g X The seminorms ∞ d kfkd,m = (`(g) + 1) kfkm g X 0 topologize S1(G; A), and since kfkm is increasing in m, the seminorms kfkm = kfkm,m also topologize S1(G; A), which is an m-algebra [Sch93, Theorem 3.1.7]. As a Frechet´ space this is just the projective tensor product of S1(G) = S1(G; C) and A. Let us denote the twistings of `1(G) (resp. S1(G)) by x `1(G; ω) (resp. S1(G; ω)). Note that the traces τ are well-deﬁned over `1(G), hence also over S1(G). ∗ Let A be a G-C -algebra, and `1(G; A) be the crossed product realized on the projective tensor G Ban product `1(G) ⊗ˆ A. If there are Dirac-dual Dirac morphisms in KK with γ = 1 in KKG (C, C), the assembly map G K∗ (EG, A) → K∗(`1(G; A)) is an isomorphism [Laf02]. For good G (including Haagerup [HK01] or hyperbolic [Laf12]), the assembly maps G K∗ (EG, A) → K∗(G nr A) ∗ is also an isomorphism, with G nr A being the reduced crossed product C -algebra. It follows that the natural map K∗(`1(G; A)) → K∗(G nr A) is an isomorphism for such groups. In the rest of the section we assume that this is the case. t Let√ω0 be an R-valued 2-cocycle on G, and consider the associated T-valued cocycles ω (g, h) = ∗ exp( −1tω0(g, h)) for t ∈ I. We then obtain a ‘continuous’ family of twisted group C -algebras C∗ (G) ∗ Γ(I C∗ (G)) r,ωt , and the C -algebra of continuous sections ; r,ω∗ plays a central role in the com- ∗ parison of their K-groups. By deﬁnition Γ(I; Cr,ω∗ (G)) is a completion of `1(G; C(I)) with respect to ∗ the representation on Hilbert C(I)-module Γ(I; `2(G)), twisted by the U(C(I))-valued 2-cocycle ω on G [ELPW10]. `1

∗ 16 Recall that Cr,ω(G) is strongly Morita equivalent to G nr K(`2(G)), where G4.1is acting-Schwartz on K(`2 algebra(G)) by ∗ (Ad (ω) )g∈G [PR89]. Concretely, the imprimitivity bimodule is given by the Hilbert Cr,ω(G)-module λg ∗ (ω) (ω) Cr,ω(G) ⊗ `2(G), where the left action of G nr K(`2(G)) is given by λg ⊗ λg for g ∈ G and 1 ⊗ T Ban for T ∈ K(`2(G)). The `1-version of this is a cycle in KK (`1(G; K(`2(G))), `1(G; ω)) implementing ∗ the isomorphism K∗(`1(G; K(`2(G)))) ' K∗(`1(G; ω)). Collecting these, `1(G; ω) → Cr,ω(G) induces isomorphisms of K-groups [Mat06].

t Proposition 4.1. The evaluation map K∗(`1(G; C(I))) → K∗(`1(G; ω )) is surjective for each t. ∗ t Proof. We already know the corresponding statement for C -algebras, and that K∗(`1(G; ω )) is isomor- K (C∗ (G)) K (` (G C(I) ⊗ K(` (G)))) ' K G C(I) ⊗ K(` (G))) phic to ∗ r,ωt . We also know that ∗ 1 ; 2 ( nr 2 . Thus, we just need to verify that the evaluation map of the assertion factors the isomorphism K∗(`1(G; C(I) ⊗ t K(`2(G)))) → K∗(`1(G; ω )) for each t. If we carry out a similar construction for `1(G; C(I)) and `1(G; C(I) ⊗ K(`2(G)), the Banach space Ban `1(G; C(I; `2(G))) gives a cycle in KK (`1(G; C(I) ⊗ K(`2(G))), `1(G; C(I))), hence a map

K∗(`1(G; C(I) ⊗ K(`2(G)))) → K∗(`1(G; C(I))).

This is the desired factorization.

Since the natural (semi)norms of S1(G; C (I)) are stronger than that of `1(G; C(I)), we may regard S (G C (I)) Γ(I C∗ (G)) 1 ; as a subspace of ; r,ω∗ ∞. In order to control the time derivatives of the product in ω ω this algebra,∞ we need to impose the polynomial growth assumption on 0. Namely, 0 is of polynomial growth if M M |ω0(g, h)| 6 C(1 + `(g)) (1 + `(h)) holds for some large enough C and M.

Proposition 4.2 (cf. [Sch93b, Theorem 6.7]). If ω0 is of polynomial growth, S1(G; C (I)) is a spectral ` (G C(I)) subalgebra of 1 ; . ∞

Proof. First, S1(G; C(I)) is a strongly spectral invariant subalgebra of `1(G; C(I)). On this part the argument for [Sch93b, Theorem 6.7] applies without change, since the twisting by ωt does not contribute to the norm of these algebras. 0 We ﬁrst need to verify that S1(G; C (I)) is a subalgebra of S1(G; C(I)). Let f and f be elements of S (G C (I)) 1 ; . Their product is represented∞ by the function

∞ t −1 0 −1 (t, g) 7→ ω (h, h g)ft(h)ft(h g).(4.2) h G X∈ t −1 t Since ω (h, h g) = 1, at each t we indeed obtain an element of S1(G; ω ). We thus need to show that this function is smooth in t. Let us denote the constants controlling the growth of ω0 as above by C and M, and put m+ = m(1 + M). As an illustration let us ﬁrst give an estimate for the ﬁrst order derivative. We have

0 ∂t(f ∗ω∗ f )t(g) t −1 −1 0 −1 0 −1 0 −1 = ω (h, h g) ω0(h, h g)ft(h)ft(h g) + (∂tft(h))ft(h g) + ft(h)∂tft(h g) .(4.3) h X 0 0 The second term can be estimated as k∂tf ∗ωt f ks,1 6 k∂tfks,1 k∂tf ks,1 as in (4.7), so

t −1 0 −1 g 7→ ω (h, h g)(∂tft(h))ft(h g) h X `1

s 17 is an element of H` (G). The third term admits a similar estimate. As for the4.1 ﬁrst-Schwartz term, we algebra need to estimate !

−1 0 −1 k ω0(h, h g) |ft(h)| ft(h g) (1 + `(g)) . g h X X M If we put f˜t(g) = ft(g)(1 + `(g)) , the above is bounded by 0 0 0 C f˜ ∗ f˜ C f˜ f˜ = C kfk f .(4.4) k,1 6 k,1 k,1 k+,1 k+,1 0 This shows that f ∗ω∗ f indeed has a bounded ﬁrst order derivative. m Let us now proceed to the higher order derivatives. For f ∈ S1(G; C (I)), put m 1 k kfkm = max ∂t fs , k! s∈I m k=0 X 0 m 0 0 k where on the right hand side we take the norm kf km = g(1 + `(g)) |f (g)| for f = ∂t fs. Given f1, ... , fn ∈ S (G; C (I)), we claim that there is an estimate 1 P

∞ f1 ··· fn enC f1 ··· kfnk 6 m+ ,(4.5) m m+ n which implies that S1(G; C (I)) is closed under product. For (g1, ... , gn) ∈ G , put

(n) ∞ ω0 (g1, ... , gn) = ω0(g1, g2) + ω0(g1g2, g3) + ··· + ω0(g1 ··· gn−1, gn). 1 n ∗ Then the product f ··· f in Γ(I; Cr,ω∗ (G)) is represented by (n) 1 n tω (g1,...,gn) ht(g) = f (g1) ··· f (gn)e 0 . g ···g =g 1 Xn k Then, ∂t hs(g) is given by

k! a (n) ∂ 1 f1(g ) ··· ∂an fn(g )ω (g , ... , g )an+1 etω0(g1,...,gn). a ··· a t s 1 t s n 0 1 n a +···+a 1! n+1! 1 X n+1 Using

(n) an+1 ω0 (g1, ... , gn) = a n+1! b1 b2 bn−1 ω0(g1, g2) ω0(g1g2, g3) ··· ω0(g1 ··· gn−1, gn) , b ! ··· b ! b + +b 1 n−1 1 ···X n−1 1 k we can bound k! maxs ∂t hs m by

Cb1+···+bn−1 a1! ··· an!b1! ··· bn−1! a1+···+an+b1 +···+Xbn−1=k

∼(bn−1) (b ) (b )∼(b2) (b ) ! (b ) a 1 a 1 a 2 a n−1 × ··· ∂^1 f1 ∂^2 f2 ∂^3 f3 ··· ∂^n fn , t s t s t t s m where we put f˜(b)(g) = f(g)(1 + `(g))Mb. Repeatedly using the estimates of the form (4.4), we can bound this by

Cb1+···+bn−1 a1 1 ∂t fs a1! ··· an!b1! ··· bn−1! m+(b1+···+bn−1)M a1+···+an+b1 +···+Xbn−1=k a × ∂ 2 f2 ··· k∂an fnk (4.6) t s t s m+bn−1M m+(b1+···+bn−1)M `2

bi C 18 Then, using b1 + ··· bn−1 6 k 6 m and i C /bi! 6 e , we obtain the desired4.2 estimate-Schwartz (4.5). algebra It remains to show that S (G; C (I)) is closed under taking multiplicative inverses. Now, suppose 1 P f ∈ S (G C (I)) S (G C(I)) f−1 that 1 ; is invertible∞ in 1 ; . The assertion follows if we can show that S (G Cm(I)) m f ∈ S (G Cm(I)) S (G C(I)) C < 1 belongs to 1 ; ∞ for all . If 1 ; is invertible in 1 ; , we ﬁx 1 and −1 0 m approximate f by f ∈ S1(G; C (I)) satisfying

C1 f0 − f−1 < , m ,0 e3C kfk + m+,0 x = 1 − f0f kxk < C e−C n > m f1 = ··· = fn = so that satisﬁes m+,0 1 . Then, for , the above estimate for x shows kxnk enCnm kxkm kxkn−m = o(Cn) m,m 6 m+,m m+,0 2 0 m for any C1 < C2 < 1. This shows f f is invertible in S1(G; C (I)). t Corollary 4.3. The evaluation map K∗(S1(G; C (I))) → K∗(S1(G; ω )) is surjective for each t.

Proof. A similar (but simpler) argument as the∞ above proposition shows that S1(G) is a spectral subalgebra of `1(G) with respect to the product ∗ωt . We thus know that the horizontal arrows in

K∗(S1(G; C (I))) / K∗(`1(G; C(I)))

∞

t t K∗(S1(G; ω )) / K∗(`1(G; ω ))

are isomorphisms. Moreover, the right vertical arrow is surjective by Proposition 4.1.

Remark 4.4. When G is of polynomial growth, there is possibly a more direct approach based on the following result of Schweitzer [Sch93b]: suppose that A is a G-C∗-algebra, and A be a dense G-invariant Frechet´ subalgebra satisfying the differential seminorm condition such that the induced action is `-tempered. Then S1(G; A) is a spectral subalgebra of G nr A. However, this result is hopeless ∗ without polynomial growth assumption. Indeed, `1(G) is not a spectral subalgebra of Cr(G) if G contains a free semigroup of two generators [Bon61, Jen70].

Remark 4.5. The subscript 1 in S1(G; A) corresponds to the fact that the deﬁnition in (4.1) is in terms ∗ of the ‘`1 norm’. When A is a pre C -algebra, the `2 version makes sense as

S (G; A) = f: G → A | ∀ k, m: α−1(f(g)∗f(g))(1 + `(g))2k < . 2  g  g  X m  ∞ This way S2(G; C) is equal to H` (G), see the next section. However, the spectral invariance is proved G A ∗ S (G ` G) only when is of polynomial growth∞ and is a commutative C -algebra [JS96], and 2 ; is a spectral subalgebra of G nr ` G (if and) only if G is of polynomial growth [CW03]. Note also that ∞ we have S1(G) = S2(G) if and only if G is of polynomial growth [Jol90]. For hyperbolic groups, there is a more elaborate choice of rapid∞ decay functions [Pus10] which are invariant under holomorphic ∗ functional calculus on Cr(G) and at the same time supporting many traces.

s > 0 4.2For -Schwartz, put algebra s 2 2s H` (G) = f: G → C | |f(g)| (`(g) + 1) < , g G X∈ T s and H` (G) = s>0 H` (G). Recall that G is said to be of rapid decay [Jol∞90] if H` (G) is a subspace C∗(G) ` Hs(G) ⊂ C∗(G) s of r ∞ for some proper length function . This is equivalent to ` r ∞for large enough . `2

∗ 19 Note that when G is ﬁnitely generated, it is rapid decay if and only if H` (G4.2) ⊂ C-Schwartzr(G) for the algebra word ` length for some symmetric generating set. Put ∞ s 2 2s kfk`,s = |f(g)| (`(g) + 1) , g G X∈ s s ∗ so that H` (G) is the completion of C[G] by this norm. When H` (G) ⊂ Cr(G), there is C > 0 such that s f B (e) e R kfk ∗ CR kfk whenever is supported on R (the ball around of radius ), we have Cr(G) 6 `,s, or that there is a polynomial P(X) such that

kf ∗ gk ∗ P(R) kfk kgk Cr(G) 6 `2(G) `2(G) whenever f is supported on BR(e). The last condition implies kf ∗ gk`,s 6 K(s) kfk`,s kgk`,s for s > deg P(X) and s P(2n)2 K(s) = 2s P(1)2 + , (1 + 2n−1)2s n X s 0 so that H` (G) is a Banach algebra with respect to the rescaled norm kfk`,s = K(s) kfk`,s [Laf00, Propo- 0 sition 1.2]. Modifying the factor K(s) if needed, we may assume that kfk`,s is increasing in s. This way ∗ H` (G) is an m-algebra. It is also closed under holomorphic functional calculus inside Cr(G). G ω T 2 G ∞Let be a rapid decay group, and be a normalized -valued -cocycle on . As explained ∗ in [Mat06, Proposition 6.11], H` (G) is also a subalgebra of Cr,ω(G) thanks to the estimate ∞ kf ∗ω gk`,s 6 k|f| ∗ |g|k`,s 6 K(s) kfk`,s kgk`,s (4.7)

for large enough s. Let us denote this algebra by H` (G; ω). It is again invariant under holomorphic C∗ (G) functional calculus in r,ω . ∞ The smooth section algebra S1(G; C (I)) needs to be replaced by C (I; H` (G)). Since C (I) is C (I H (G)) C (I) a nuclear Frechet´ space, ; ` ∞which is naturally the injective∞ tensor∞ product of ∞ and H (G) ` is also the projective∞ tensor∞ product. ∞ ∞Then, an analogue of the ﬁrst half of Proposition 4.2 holds.

Proposition 4.6. Suppose that ω0 is of polynomial growth. Then C (I; H` (G)) is an m-convex subalgebra Γ(I C∗ (G)) of ; r,ω∗ . ∞ ∞ 0 ∗ Proof. Let f and f be elements of C (I; H` (G)). Their product in Γ(I; Cr,ω∗ (G)) is represented by t H (G ωt) (4.2), and by the estimate (4.7), at each∞ we∞ indeed obtain an element of ` ; . We thus need to t show that this function is smooth in . ∞ The proof is completely analogous to the ﬁrst half of Proposition 4.2, with slight modiﬁcations using (4.7). For example, in (4.3) the second term can be estimated as

0 0 ∂ f ∗ t f K(s) k∂ fk ∂ f t ω `,s 6 t `,s t `,s as in (4.7), and the third term admits a similar estimate. As for the ﬁrst term, we need to estimate

−1 0 −1 2s ω0(h, h g) |ft(h)| ft(h g) (1 + `(g)) . g h X X M If we put f˜t(g) = ft(g)(1 + `(g)) , the above is bounded by

0 2 2 0 2 2 0 2 C f˜ ∗ f˜ 6 CK(s) f˜ f˜ = CK(s) kfk M f M .(4.8) `,s `,s `,s `,s+ 2 `,s+ 2

0 This shows that f ∗ω∗ f indeed has a bounded ﬁrst order derivative. `2

20 Let us now proceed to the higher order derivatives. For f ∈ C (I; H` (G)), again4.2 -Schwartz put algebra m 1 ∞ ∞ k kfkm = max ∂t fs . k! s∈I `,m k=0 X 1 n Let C and M be the constants controlling the growth of ω0. When f , ... , f in C (I; H` (G)), instead of (4.5) we claim n ∞ ∞ f1 ··· fn K(m )eC f1 ··· kfnk 6 + m+ (4.9) m m+ M n with m+ = m(1 + 2 ) this time. For (g1, ... , gn) ∈ G , put (n) ω0 (g1, ... , gn) = ω0(g1, g2) + ω0(g1g2, g3) + ··· + ω0(g1 ··· gn−1, gn). 1 n ∗ 1 k h = f ··· f Γ(I C ∗ (G)) ∂ h Then, with the product in ; r,ω , we can bound k! maxs t s `,m by

K(m + (b + ··· + b ) M )nCb1+···+bn−1 1 n−1 2 ∂a1 f1 t s M a1! ··· an!b1! ··· bn−1! `,m+(b1+···+bn−1) 2 a1+···+an+b1 +···+Xbn−1=k

a2 2 an n × ∂t fs ··· k∂t fs k M (4.10) M `,m+bn−1 2 `,m+(b1+···+bn−1) 2

bi repeatedly using the estimates of the form (4.8). Then, using b1 + ··· bn−1 6 k 6 m and i C /bi! 6 eC, we obtain the desired estimate (4.9). P The rest proceeds as in the proof of [MRZ˙ 61, Lemma 1.2]. Namely, having the m-convexity is equivalent to ﬁnding a system of convex neighborhoods of 0 which are idempotent U · U ⊂ U. Putting Bm(r) = {f: kfkm < r} ⊂ C (I; H` (G)), our claim above implies n ∞ ∞ −1 −C B M K(m+) e ⊂ Bm(1) m(1+ 2 ) −1 −C n for any n. Thus, the convex hull Um of the Bm+ (K(m+) e ) for n = 1, 2, ... is a neighborhood 2 1 0 of 0 satisfying Um = Um. Then m0 Um for m, m = 1, 2, ... is a fundamental system of idempotent neighborhoods.

Proposition 4.7. Under the same assumption as Proposition 4.6, C (I; H` (G)) is a spectral subalgebra of Γ(I C∗ (G)) ; r,ω∗ . ∞ ∞ m Proof. First we claim that C(I, H` (G)), or equivalently C(I, H` (G)) when m is sufficiently large, is Γ(I C∗ (G)) a spectral subalgebra of ; r,ω∞∗ . The proof for each fiber [Mat06, Proposition 6.11] carries on ∗ to this case, by replacing the norm of `2(G) by that of the Hilbert Γ(I; Cr,ω∗ (G))-module C(I; `2(G)). (Otherwise we can appeal to [Par13].) The rest of proof is analogous to the second half of Proposition 4.2. Suppose that f ∈ C (I; H` (G)) C(I H (G)) f−1 Cm(I Hm(G)) is invertible in , ` . The assertion follows if we can show that belongs to ∞ ; `∞ m f0 ∈ Cm(I Hm(G)) when is sufficiently∞ large. Let us take an element , ` satisfying −1 −1 0 C f − f < C K (m ) e kfk m+ m+ 1 + C(I,H (G)) C(I,H` (G)) ` −1 −C 0 for some 0 < C1 < 1. Thus, we have kxk m+ < C1K (m+) e for x = 1 − f f. Using this C(I,H` (G)) n n −1 we are going to show that kx km = o(C2 ) for any C1 < C2 < 1, which implies that (1 − x) , hence f−1 = (1 − x)−1f0 also, is in Cm(I, Hm(G)). ` 1 n 1 k n Replacing f , ... , f in the proof of Proposition 4.6 by x, from (4.10) we can bound k max ∂t xs ! s∈I `,m from above by K(m )nenC + ∂a1 x ··· k∂an x k t s `,m t s `,m+ . a ! ··· an! + a + +a k 1 1 ···X n6 21 Now, let us suppose n > m so that the ai start to contain 0. f we fix the possibility4.3 Effective ofconjugacy the possible problem values of a1, ... , an up to permutation, the multiplicity in the above sum is bounded by n(n − 1) ··· (n − k + m 1) 6 n . Hence we have n n nC m m n−m kx k K(m ) e n kxk kxk m m 6 + m+ + . C(I,H` (G)) n By assumption on kxk m+ , we indeed have that this is asymptotically bounded by C2 as C(I,H` (G)) n → .

∞

φ A = S (G) H (G) We4.3 nowEffective want conjugacy to use the problem formula of Proposition 2.4. Let be a cyclic cocycle on 1 or ` x ξ φ(t) supported on the conjugacy class of . As follows from the precise form of given in (3.1), indeed∞ t t extends to a cyclic cocycle on Aωt = S1(G; ω ) or H` (G; ω ) for each t. ξ(g ··· g ) `(g ) ... `(g ) (φ(t)) If the growth of 0 n is bounded by some∞ polynomial in 0 , , n , the family t is smooth in t and θ in Section 3 makes sense as a functional on A ⊗ˆ A. Since the growth ω0(g, h) is bounded by some polynomial in `(g) and `(h), it is enough to know that `(gg0 ) is bounded by some polynomial in `(g0) when g0 is conjugate to x. Note also that if CG(x) is finite, the growth rate does not depend on the choice of gg0 . Such problem (usually without finiteness assumption on the order of x) is known as the effective conjugacy problem: consider the conjugacy length function

CLFG(n) = max min `(g). x, y conjugate gxg−1=y `(x)+`(y)=n

This is known to be at most linear for the hyperbolic groups [Lys89], for the solvable groups of the form k Z oA Z with A ∈ SLk(Z) contained in a split torus [Sal16], or for the torsion free 2-step nilpotent groups [JOR10], as well as the crystallographic groups (see the next section). A quadratic bound is known for fundamental groups of prime 3-manifolds [BD14], see also [Sal16, Theorem 5.5]. Recall that, in the first half of the next theorem, G being ‘good’ means that it satisfies the Baum– Connes conjecture with coefficients and that the natural map K∗(`1(G; A)) → K∗(G nr A) is isomorphism for any G-C∗-algebra A, which are satisfied by the Haagerup groups and the hyperbolic groups.

Theorem 4.8. Let G be a good group with polynomially bounded conjugacy length function, and ω0 be an R-valued 2-cocycle of polynomial growth. Furthermore, let x be a torsion element of G such that et[ω0] makes ∗ sense in H (CG(x); C). Then for any cyclic cocycle φ on S1(G) supported on the conjugacy class of x and periodic cyclic cycle c ∈ HP∗(S1(G; C (I))), the pairing

∞ (t) t[ω ] hevt c, φ ∪ e 0 i is constant in t. An analogous statement holds for cyclic cocycles on H` (G) when G is of rapid decay. (t) −t[ω ] ∞ We thus have hevt c, φ i = hev0 c, φ ∪ e 0 i in the above setting. Note also that, as we have K S (G C (I)) C∗ (G) seen in the previous section, the -groups of 1 ; and r,ωt are naturally isomorphic, so ∗(S (G ωt)) K (C∗ (G)) this gives a concrete description of the pairing between∞ HC 1 , and ∗ r,ωt in terms of ∗ ∗ the one between HC (S1(G)) and K∗(Cr(G)).

Let us consider a discrete G sitting in an exact sequence 1 → Zn → G → F → 1 such that F is 5finite, k-theory and its induced of action twisted on Zn is free crystallographic away from origin. These groupgroups are algebras of polynomial growth, hence are amenable and there is no distinctionk-theory between of twisted full crystallographicand reduced twisted group groups algebras algebras. 22 We also know that various classes of rapid decay functions coincide [Jol90], so we just write S(G). By [Sal12, Corollary 2.3.18] the conjugacy length function of G has a linear growth. Moreover, one may take Rn as a model of EG (universal space of the proper G-actions) [CK90], hence G\Rn as a one for BG = G\EG. Any R-valued 2-cocycle is up to cohomology represented by an F-invariant 2-cocycle on Zn. n Example 5.1. Let G be the group Z . For each n × n real√ skew-symmetric matrix θ, we can construct a 2-cocycle on this group by defining ωθ(x, y) = exp( −1hθx, yi). The corresponding twisted group ∗ ∗ n C -algebra C (G, ωθ) is also called the noncommutative torus C(Tθ ). For n = 2, since θ depends on θ0 only one number as hθx, yi = 2 (x2y1 − x1y2), we call that number θ also. Example 5.2. Let F be a finite subgroup of GLn(Z) such that each W ∈ F leaves θ invariant, i.e., T 0 n 0 W θW = θ. Then we can define a 2-cocycle ωθ on G = Z o F by ωθ((x, g), (y, h)) = ωθ(x, g.y) for x, y ∈ Zn and g, h ∈ F. If each W ∈ F not identity acts free away from origin, we obtain a crystallographic group. Again note that when n = 2, any finite subgroup of SL2(Z) will do. Let M denote the set of conjugacy classes of maximal finite subgroups of G. In the following theorem, K˜ 0(C[P]) denotes the reduced K0-group, that is, the kernel of the map K0(C[P]) → K0(C) induced by ∗ n ∗ the trivial representation. Note also that the map Ki(C (Z )) → Ki(C (G)) induced by the inclusion n ∗ n Z → G factors through the space of coinvariants Z ⊗F Ki(C (Z )).

Theorem 5.3 ([LL12]). In the even degree, there exists an exact sequence

M ∗ 0 K˜ 0(C[P]) K0(C (G)) K0(BG) 0, (5.1) P∈M which splits after inverting |F|. The natural map ∗ n M ∗ Z ⊗F K0(C (Z )) ⊕ K˜ 0(C[P]) → K0(C (G)) P∈M

∗ also becomes a bijection after inverting |F|. In the odd degree, we have K1(C (G)) ' K1(BG).

Under the above setting, any nontrivial finite subgroup of G has a finite normalizer [LS00, Lemma 6.1]. In particular, any nontrivial torsion element has a finite centralizer. Thus, for the twisted group algebras, the pairing with cyclic cocycles supported by the conjugacy class of such an elements is described by Section 3.1. As for the factor labeled by the identity element, For the above examples H∗(G; C) is bounded in degree by n. Hence the assumptions of Theorem 4.8 (t) are satisfied and we can compute the pairings hevt c, φ i for c ∈ HP∗(S1(G; C (I))). For x = e, the K∗(BZn) ' K∗(Tn) ∗(C (Tn)) ' L H (Tn C) pairing reduces to that between and HP ∞k ∗+2k ; with et[ω0] [ω ] ∈ H2(Tn C) x τx modification by with 0 ; , while for nontrivial∞ torsion , the pairing with ωt essentially does not see t. ∗ n for the even degree, on the image of K0(C (Z )) it reduces to the pairing between

2∗ ^ F Cn ' H2∗(Zn; C)F ' H2∗(BZn; C)F

∗ n n and Z ⊗F K0(C (Z )) ' Z ⊗F H2∗(BZ ; Z). On the image of K˜ 0(C[P]), we just pick up the coefﬁcient of e ∈ G in the projections representing the K˜ 0-classes. For the odd degree it reduces to the natural V2∗+1 n F 2∗+1 n F pairing of ( C ) ' H (Z ; C) and K1(BG) ' K1(BG). ∗ n ∗ If F = Zm, the kernel of natural map Z ⊗F K1(C (Z )) → K1(C (G)) is annihilated by the mul- tiplication by m, the K-groups of C∗(G) are free commutative groups whose rank can be explicitly described [LL12]. Let us further assume that p = m is prime. Then any non-trivial ﬁnite subgroup P 2 Z Z3

23 of G must be isomorphic to Zp via the restriction of the projection map5.1 CrossedG → Zp product. Thus any of nontrivialby ﬁnite subgroup of G represents an element of M. We have the following precise description of the K-groups. 0 Theorem 5.4 ([DL13]). The number n0 = n/(p − 1) is an integer, and |M| = pn . The rank of K-homology groups of BG are given by

0 0 2n + p − 1 (p − 1)pn −1 2n + p − 1 (p − 1)pn −1 rk K (BG) = + , rk K (BG) = − . 0 2p 2 1 2p 2 ˜ ∗ ∗ 0 We want to explain how projections in K0(C[P]) contribute to the K0(C (G) and K0(C (G, ωθ)). p Since C[P] is isomorphic to the algebra C(Pˆ ) ' C , K0(C[P]) is the free abelian group of rank p. Now suppose g is a generator of P. The minimal projections of C[P] are given by

p−1 1 √ 2π Q = exp −1 jk λ k j,g p p g k=0 X for j = 0, ··· , p − 1, which also represent a basis of K0(C[P]). Since Q0,g represents the trivial representation, a basis of K˜ 0(C[P]) is given by Q1,g, ··· , Qp−1,g. ∗ By the above theorem, these projections are still linearly independent in K0(C (G)). To get elements ∗ 0 of K0(C (G, ωθ)), we need to modify this presentation a bit. Continuing to denote a generator of ∗ ∗ 0 P by g the unitary λg has order p in C (G). But since we modiﬁed the product in C (G, ωθ), the 0 (ωθ) 0 unitary λg need not be so. Still, since ωθ is cohomologically trivial on the ﬁnite group P and hence 0 (ωθ) C 0 [P] ' C[P], we can always multiply suitable z ∈ T so that order of the unitary zλg is p. Then a ωθ similar formula p−1 (θ) 1 √ 2π (ω0 ) Q = exp −1 jk zkλ θ j,g p p gk k=0 X ∗ 0 for j = 1, ··· , p − 1 will give projections which are elements of K0(C (G, ωθ)). n Remark 5.5. Yashinski [Yas13] proposed another approach to compare HP∗(S(Z , ωθ)) for different θ based on A -isomorphism of Cartan–Eilenberg complexes, which is equivariant under the action of Zm as above and would provide alternative route to the above result. While this is an interesting viewpoint, our understanding∞ is that there is a gap in his argument, mainly because the cyclic bicomplex is not functorial for A -morphisms.

∞ 2 Z Z3

2 5.1Let usCrossed come back product to Example of by2.3. This group Z o Z3 can be presented as D E G = u, v, w | w3 = e, uv = vu, wuw−1 = u−1v, wvw−1 = u−1 .

Up to conjugation, the finite subgroups of G are generated by one of the elements w, uw or, u2w all 0 of which are of order 3. Since ωθ(w, w) = 1, the first element still has order 3 in the twisted group (θ) (θ) ∗ 0 algebra, and the corresponding projections Q1,w and Q2,w give nontrivial classes K0(C (G, ωθ)). On the other hand, √ 0 0 −1 −1 − −1 θ ωθ(uw, uw)ωθ(uwuw, uw) = ωθ(u, u v)ωθ(v, v ) = e 2 0 √ 0 (ωθ) θ (ωθ) shows that λuw is not of order 3. But an adjustment like exp( −1 6 )λuw gives the right order. When x, g ∈ G are torsion elements and the intersection AdG(x) ∩ hgi is nontrivial, there is a nontrivial (θ) x pairing between Q and τ 0 . Theorem 3.4 says that, due to an intricate way the coefficients in both j,g ωθ x (θ) are modified by θ, the value of τ 0 (Q ) remains constant. Let us compare this to [BW07]. ωθ j,g √ (ω0 ) REFERENCES1 θ 24 For simplicity, let us write e(t) instead of exp( −1tθ). Then as noticed above e( 6 )λuw is an (ω0 ) e( 2 )λ θ unitary of order 3. Similarly one can see that 3 u2w is also an unitary of order 3. Then we (θ) (θ) (θ) (θ) (θ) (θ) have the following projections Q , Q , Q , Q , Q , Q defining nontrivial classes in 1,w 2,w 1,uw 2,uw 1,u2w 2,u2t ∗ 0 K0(C (G, ωθ)). Recall from [BW07] that if α defines an action of Zp on A, an α-invariant functional φ on A is said to be an α-trace if φ(xy) = φ(α(y)x) s holds for any x, y ∈ A. An α -trace φ gives rise to a trace Tφ on Zp n A defined by

p−1 Tφ(x0 + x1w + ··· + xp−1w ) = φ(xp−s)

for xi ∈ A and w being the copy of 1 ∈ Zp. 2 Coming back to the example, let us consider the induced action of Z3 on S(Z , ωθ). From [BW07, 1 2 Theorem 3.3], we have following α-traces φl on S(Z , ωθ): 1 φ1(λ(ωθ)mλ(ωθ)n) = e ((m − n)2 − l2) δ (m − n − l) l u v 6 3Z

2 2 2 for l = 0, 1, 2. Similarly we have following α -traces φl on S(Z , ωθ): 1 φ2(λ(ωθ)mλ(ωθ)n) = e −mn − ((m − n)2 − l2) δ (m − n − l) l u v 6 3Z

i i 2 0 for l = 0, 1, 2. Each φ then gives the the traces T = T i on Z3 n S(Z , ωθ) = S(G, ω ). Using l l φl θ (ωθ)m (ωθ)n mn (ωθ) t λu λv = e(− )λumvn , it is straightforward to check that the trace τ 0 of Example 2.3 is 2 ωθ 2 exactly equal to T0 . The pairing of the above projections and traces are given by the following table (cf. [BW07, Theorem 1.2]), which is independent of θ as suggested by Theorem 3.4.

w2 1 uw2 1 u2w2 1 w 2 uw 2 u2w 2 τω0 = T0 τω0 = T1 τω0 = T2 τω0 = T0 τω0 = T1 τω0 = T2 θ √ θ θ θ √ θ θ (θ) 1 4π −1 1 2π −1 Q e 3 0 0 e 3 0 0 1,w 3 √ 3 √ (θ) 1 2π −1 1 4π −1 Q e 3 0 0 e 3 0 0 2,w 3 √ 3 √ (θ) 1 4π −1 1 2π −1 Q 0 0 e 3 0 e 3 0 1,uw 3 √ 3 √ (θ) 1 2π −1 1 4π −1 Q 0 0 e 3 0 e 3 0 2,uw √ 3 3 √ (θ) 4π 2π 1 3 −1 1 3 −1 Q 2 0 e 0 0 0 e 1,u w 3 √ 3 √ (θ) 1 2π −1 1 4π −1 Q 0 e 3 0 0 0 e 3 2,u2w 3 3

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